Understanding Vector Spaces, Linear Independence, Bases, and Dimension

Introduction

This article reviews the fundamental concepts of vector spaces, how to determine whether a set is a vector space, and the related notions of linear combinations, generating families, linear independence, bases, and dimension. The goal is to give you a complete picture so you no longer need to watch the original video.

Definition of a Vector Space

• A vector space (E) is a set equipped with two operations:
• Addition (internal law): for any (x, y \in E), (x + y \in E).
• Scalar multiplication (external law): for any scalar (\lambda) from a field (\mathbb{K}) (usually (\mathbb{R}) or (\mathbb{C})) and any (x \in E), (\lambda x \in E).
• The addition must make (E) an abelian group (associative, commutative, neutral element (0), and each element has an opposite (-x)).
• Scalar multiplication must satisfy the usual distributive and associative properties (e.g., (\lambda(x+y)=\lambda x+\lambda y), ((\lambda+\mu)x=\lambda x+\mu x), ((\lambda\mu)x=\lambda(\mu x)), and (1x=x)).

Subspaces

A subset (F \subseteq E) is a subspace if: 1. (F) is non‑empty (usually shown by proving (0 \in F)). 2. (F) is closed under addition: (x, y \in F \Rightarrow x+y \in F). 3. (F) is closed under scalar multiplication: (\lambda \in \mathbb{K}, x \in F \Rightarrow \lambda x \in F). If these three conditions hold, (F) inherits all vector‑space properties from (E).

How to Prove (or Disprove) That a Set Is a Vector Space

1. Identify a known ambient space (e.g., (\mathbb{R}^n), (\mathbb{C}^n), polynomial spaces, matrix spaces). Show that your set is a subset of this space.
2. Check the three subspace conditions listed above. In practice, the group axioms for addition are automatically satisfied because they hold in the ambient space.
3. Look for the neutral element: if the zero vector is missing, the set cannot be a vector space.
4. Use alternative criteria when convenient:
5. Show the set is the kernel or image of a linear map.
6. Show the set can be written as a span of known vectors (then it is automatically a subspace).

Classical Examples of Vector Spaces

• (\mathbb{R}^n) and (\mathbb{C}^n).
• The space of all real (or complex) polynomials (\mathbb{R}[X]) or (\mathbb{C}[X]).
• The subspace (\mathbb{R}_n[X]) of polynomials of degree ≤ n.
• Spaces of matrices of a fixed size, e.g., (M_{m\times n}(\mathbb{R})).
• Spaces of sequences, continuous functions, (C^1) functions, etc. Each of these satisfies the axioms automatically; the main work is to verify that a new set is a subspace of one of them.

Linear Combinations

Given vectors (u_1,\dots,u_k) in a vector space, a linear combination is any vector of the form [\sum_{i=1}^{k}\lambda_i u_i] with scalars (\lambda_i \in \mathbb{K}). The set of all linear combinations of a family ({u_i}) is called its span and is denoted (\operatorname{span}{u_i}).

Generating Families

A family ({u_i}_{i\in I}) generates a vector space (E) if (\operatorname{span}{u_i}=E). In geometric terms: - One non‑zero vector generates a line through the origin. - Two non‑collinear vectors generate a plane. - Three non‑coplanar vectors generate the whole three‑dimensional space, and so on. If a family contains vectors that are already linear combinations of the others, those vectors are redundant and can be removed without changing the span.

Linear Independence (Free Families)

A family ({u_i}{i\in I}) is linearly independent (or free) if the only linear combination that yields the zero vector is the trivial one: [\sum{i\in I}\lambda_i u_i = 0 \;\Longrightarrow\; \lambda_i = 0 \text{ for all } i.] Equivalently, no vector in the family can be expressed as a linear combination of the others. To prove independence, assume a linear relation equals zero and show that each coefficient must vanish (often by comparing coordinates or using known properties of the vectors).

Bases

A basis of a vector space (E) is a family that is both: 1. Generating (its span equals (E)). 2. Linearly independent. When a basis exists, every vector of (E) can be written uniquely as a linear combination of the basis vectors. The most familiar bases are the canonical bases: - For (\mathbb{R}^n): (e_1=(1,0,\dots,0),\; e_2=(0,1,0,\dots,0),\dots, e_n). - For (\mathbb{R}_n[X]): (1, X, X^2,\dots, X^n). Any other basis can be obtained by applying an invertible linear transformation to a canonical basis.

Dimension

The dimension of a vector space (E) is the number of vectors in any basis of (E). Important facts: - All bases of a given space have the same cardinality. - In a space of dimension (n): * No independent family can contain more than (n) vectors. * Any generating family must contain at least (n) vectors. * An independent family of exactly (n) vectors is automatically a basis, and a generating family of exactly (n) vectors is also a basis. - To find the dimension, one often constructs a maximal independent set (or a minimal generating set) and counts its elements.

Practical Tips for Exercises

1. Start from the definition: verify the three subspace conditions.
2. Look for the zero vector early; its absence disproves the vector‑space property instantly.
3. Use spans: if you can write the set as (\operatorname{span}{v_1,\dots,v_k}) for known vectors, you are done.
4. Check independence by solving the homogeneous linear system (\sum \lambda_i v_i = 0).
5. Determine a basis by discarding redundant vectors from a generating family or by extending an independent family until it spans the whole space.
6. Remember the dimension: it tells you the exact size a basis must have, which simplifies many proofs.

Summary of Methods to Prove a Set Is a Basis

Closing Remarks

Understanding vector spaces is a matter of mastering three intertwined ideas: closure (subspace criteria), combination (spans), and independence (bases). Once these are clear, the rest of linear algebra—linear maps, matrices, eigenvalues—becomes much easier.

A set is a vector space precisely when it contains the zero vector and is closed under addition and scalar multiplication; a basis is a minimal generating family that is also linearly independent, and its size defines the dimension of the space. Mastering these concepts lets you work with any vector space without needing the original video.